JEE - Algebra

Algebra is a mathematical topic that covers number theory, geometry, and analysis. In the history of Mathematics, it is one of the first branches. According to the definition of algebra, the study of mathematical symbols and rules sometimes includes manipulating these mathematical symbols. From solving simple equations to analysing abstractions, algebra covers a wide range of topics. Many chapters of Mathematics that kids will study in school contain algebra equations. Algebra also has a number of formulae and identities.

Important Terms in Algebra

• Polynomials

• Quadratic Equation

• Domain and Range

• Trigonometric function

• Exponential function 

• Irrational expression

• Inequalities

• Linear equations

• Sequences and series 

What is Algebra

We use mathematical formulae to determine the distance and capacity of containers in our daily lives, as well as to compute sales pricing when necessary. Algebra is a mathematical language that allows you to express a mathematical equation and connection using letters or other symbols to represent things. The unknown quantities in the equation can be solved using algebra.

Algebra helps in the solution of mathematical equations and the calculation of unknown numbers such as bank interest, proportions, and percentages. We can rewrite the equations by using the variables in algebra to represent the unknown quantities that are related.

Classification of Algebra 

The branches and sub-branches of Algebra are listed below. These branches are used to find the values of two or more variables in the correct order. The following are the algebraic branches:

Elementary Algebra

The conventional topics studied in a modern elementary algebra course are covered in Elementary Algebra. Arithmetic is the study of numbers and mathematical operations such as +, -, x, and y. However, in algebra, numbers are frequently represented by symbols and are referred to as variables, such as x, a, n, and y. It also allows for the common formulation of arithmetic laws such as a + b = b + a, and it is the first step in demonstrating the systematic exploration of all properties of a system of real numbers.

Variables, evaluating expressions and equations, properties of equalities and inequalities, solving algebraic equations and linear equations with one or two variables, and other concepts are covered in elementary algebra.

Advanced Algebra

This is Algebra's intermediate level. In comparison to pre-algebra, this algebra has a lot of equations to solve. Advanced algebra will help you in understanding other topics of algebra, such as:

• Equations with inequalities

• Matrices

• Solving system of linear equations

• Graphing of functions and linear equations

• Conic sections

• Polynomial equation

• Quadratic functions with inequalities

• Polynomials and expressions with radicals

• Sequences and series

• Rational expressions

• Trigonometry

• Discrete Mathematics and probability

Abstract Algebra

Abstract algebra is one of the divisions of algebra that tries to solve truths regarding algebraic systems that are independent of the nature of particular operations. In some circumstances, these operations have particular features. As a result, abstract algebra is a branch of Mathematics.

The concepts of abstract algebra are as follows:

1. Sets – Sets are defined as a collection of items whose existence is determined by a set's special property. For example, a collection of all 22 matrices, a collection of two-dimensional vectors in the plane, and many types of finite groups.

2. Binary Operations – The binary operations are created when the concept of addition is realised. Without a set, the concept of all binary operations will be useless.

3. Identity Element – The digits 0 and 1 are used to represent an identity element for a certain operation. For the addition operation, 0 is the identity element, but for the multiplication operation, 1 is the identity element.

4. Inverse Elements – The idea of inverse elements comes up with a negative number. For addition, we write “-a” as the inverse of “a” and for the multiplication, the inverse form is written as “a-1″.

5. Associativity – When adding integers, there is a property known as associativity, which states that grouping the numbers added has no effect on the result. Consider an example, (5 + 2) + 7 = 5 + (2 + 7)

Linear Algebra

Linear algebra is an area of Mathematics that may be used in both practical and pure Mathematics. It is concerned with linear mappings between vector spaces. It also entails the investigation of planes and lines. It is the study of transformation qualities in linear sets of equations. It is nearly universally used in Mathematics. It is concerned with linear equations and their representation in vector spaces and matrices for linear functions.

The following are some of the most essential subjects studied in linear algebra:

• Linear equations

• Vector Spaces

• Relations

• Matrices and matrix decomposition

• Relations and Computations

Commutative Algebra

Commutative algebra is one of the branches of algebra that studies the commutative rings and their ideals. The algebraic number theory, as well as the algebraic geometry, depends on commutative algebra. Rings of algebraic integers, polynomial rings, and so on are examples. Commutative algebra is used in many other areas of Mathematics, including differential topology, invariant theory, order theory, and general topology. In modern pure Mathematics, it has played a significant role.

Solved Examples

Example 1: Solve $\left|x^{2}+x-4 \right|= \left|x^{2}-4 \right|+\left|x \right|$

Solution: $\left|x^{2}+x-4 \right|= \left|x^{2}-4 \right|+\left|x \right|$

$\Rightarrow x\left ( x^{2}-4 \right )\geq 0$

$\Rightarrow x\left ( x-2 \right )\left ( x+2 \right )\geq 0$

$x\in \left [ -2, 0 \right ]\cup \left [ 2, \infty  \right ]$

Example 2: Find how many of the equations $x^{4}+2x^{2}-8x+3=0$ are real. 

Solution: Let $f(x)= x^{4}+2x^{2}-8x+3=0$

$\Rightarrow f'(x)= 4x^{3}+4x-8$

$\Rightarrow f'(x)= 4x^{3}+4x-8= 4(x-1)\left ( x^{2x}+4x-8 \right )$

now , $f'(x)=0\Rightarrow x =1$

Hence, the graph of y = f(x) has only one turn (maxima/minima)

Now, $f(1)=1+2-8+3<0$

Also when $x\to \pm \infty , f(x) \to \infty$

Solved Problems of Previous Year Question

1. A value of b for which the equations $x^{2}+bx-1=0,\,\  x^{2}+x+b=0$  have one root in common is 

1. $-\sqrt{2}$

2. $-i\sqrt{3}$

3. $\sqrt{2}$

4. $\sqrt{3}$

Ans: (b) here, we have

$x^{2}+bx-1=0$ ……… (i)

$x^{2}+x+b=0$ …………(ii)

Subtracting, we get $\left ( b-1 \right )x-1-b=0$

$\Rightarrow x=\frac{b+1}{b-1}$ 

This value of x satisfies equation (1)

$\Rightarrow \frac{(b+1)^{2}}{(b-1)^{2}} +\frac{b+1}{b-1}+b=0$

$\Rightarrow b=\sqrt{3}i, \,b=-\sqrt{3}i,\, 0$  

2. Let $\alpha$ and $\beta$ be the roots of $x^{2}-6x-2=0, \,\ with ,\ ,\alpha > \beta$. If $a_{n}=\alpha ^{n}-\beta ^{n}$ for $n\geq 1,$ then the value of $\frac{a_{20}-2a_{8}}{2a_{9}}$. 

1. 1

2. 2 

3. 3

4. 4

Answer: c

$a_{n}=\alpha ^{n}-\beta ^{n}$

Also $\alpha ^{2}-6\alpha -2=0$

Multiply with $\alpha^{8}$  on both sides

$\Rightarrow \alpha ^{10}-6\alpha ^{9}-2\alpha ^{8}=0$ ……….. (i)

Similarly $\beta^{10}-6\beta ^{9}-2\beta ^{8}=0$ ………. (ii)

Subtracting (2) from (1) we have 

$\Rightarrow \alpha ^{10}-\beta ^{10}-6\left ( \alpha ^{9}-\beta ^{9} \right )=2\left ( \alpha ^{8}-\beta ^{8} \right )$

$\Rightarrow a_{10}-6a_{9}=2a_{8}$

$\frac{a_{20}-2a_{8}}{2a_{9}}=3$

3. The number of distinct real roots of $x^{4}-4x^{3}+12x^{2}+x-1 = 0$ is ……

Ans: b

Let $f(x) = x^{4}-4x^{3}+12x^{2}+x-1 = 0$

$\therefore f'(x)= 4x^{3}-12x^{2}+24x+1$

And $f''(x)=12x^{2}-24x+24>0$ for all real x.

So, the graph of f’(x) intersects the x - axis only once.

Hence, f(x) has only one turning point.

Also, f(0)= -1.

So, the graph of f(x) cuts the x axis at two points.

Hence, f(x) = 0 has two real roots.

Practice Question:

1. The sum of the non - real roots of $\left ( x^{2}+x-2 \right )\left ( x^{2} +x - 3\right )=12$ is 

1. -1

2. 1

3. -6

4. 6

Answer:  (a)

2. The number of irrational roots of the equation $\frac{4x}{x^{2}+x+3}-\frac{5x}{x^{2}-5x+3}=- \frac{3}{2}$

1. 4

2. 0 

3. 1

4. 2

Answer: (d)

Conclusion

In this chapter, we have elaborated on concepts and solutions to questions on the topic Algebra. Everything you're looking for is available in a single location. Students can carefully read through the concepts, Definitions, and questions in the PDFs, which are also free to download and understand the concepts used to solve these questions. This will be extremely beneficial to the students in their exams.